Chapter 1 (Basic Probability)
|
|
- Derek Hoover
- 6 years ago
- Views:
Transcription
1 Chapter 1 (Basic Probability) What is probability? Consider the following experiments: 1. Count the number of arrival requests to a web server in a day. 2. Determine the execution time of a program. 3. No of calls received at the IR help desk. What is common to the above experiments? Random phenomena in computer environments: Arrival of jobs. Execution time. Memory requirement. Failure of components. Exposure to viruses. Errors in codes. Etc. Random phenomena elsewhere: Economy: Stock prices, number of jobs, price of oil, etc. Environment: Temperature, earthquakes, rainfall, etc. Going to UTD: # of green lights, available parking, etc. This class: Scores, time spent on each topic, etc. This course: Model uncertainty, quantify uncertainty and make decisions under uncertainty. Probability means Chance (common language) Odds (gambling) Long-term proportion (relative frequency) Likelihood (forecasting) Finite measure (mathematics) Probability = function of an event = P(E) What is an event? Consider a random experiment. Its results are outcomes. Sample space S = {all possible outcomes} Event E = a combination of outcomes = a subset of S Event E occurs when the experiment results in an outcome that is contained in E. 1
2 Range of probability: Ex: Find S for the experiments 1-3 mentioned at the beginning. Ex: Consider the execution of an if statement twice: if then T else E. What is S? Ex: Consider a system with two components. The experiment consists of observing the status of the components: W (working) or F (failed). Describe the following events. (a) At least one component has failed. (b) Exactly one component has failed. Finding probability of an event E: If E = {O 1, O 2,, O k } = {outcomes}, then P(E) = An empty event φ = {} has P(φ) = 0. P(S) = Some set operations: E 1, E 2,, E n events (sets of outcomes) Union of events E 1, E 2,, E n is an event that consists of all outcomes of E 1, E 2,, E n occurs if any of E 1, E 2,, E n occurs (or at least one of E 1, E 2,, E n occurs). is denoted as E 1 E 2 E n = {E 1 or E 2 or or E n } Venn Diagram: 2
3 Intersection of events E 1, E 2,, E n is an event that consists of common outcomes of E 1, E 2,, E n occurs if each E 1, E 2,, E n occurs is denoted as E 1 E 2 E n = {E 1 and E 2 and and E n } Venn Diagram: Complement of an event E is an event that consists of outcomes that are not in E occurs if E does not occur is denoted as E C = {not E}. Venn Diagram: What is the Venn Diagram for the event A C B? Disjoint or mutually exclusive events are those that cannot occur together their intersection = {} Venn diagram: Exhaustive events are those whose union = Sample space at least occurs for sure. Example =? 3
4 Basic rules of probability For any event E, P({}) = and P(S) = For disjoint events E 1, E 2,, E n : P(E 1 E 2 E n ) = For any two events E 1 and E 2 : P(E 1 E 2 ) = Complement rule. For event E and its complement E C : P(E E C ) = Which one should we compute, P(E) or P(E C )? Independent events: P(E 1 E 2 ) = P(E 1 ) P(E 2 ) P(E 1 E 2 E n ) = P(E 1 ) P(E 2 ) P(E n ) Basic idea: One event has information about the other event. Q. Can disjoint events be independent? Q. Can exhaustive events be independent? (More on independence later.) How to find probability of an event? Recall: If E = {O 1, O 2,, O k } = {outcomes}, then P(E) = Q: How to assign probabilities to the outcomes? 4
5 Classical approach for assigning probability: Assume a mathematical model for the outcomes in S. Popular model: When S = finite = {O 1, O 2,, O N }, assume that the outcomes {O i } are equally likely, i.e., P(O 1 ) = P(O 2 ) = = P(O N ) = Then, for any event E, P(E) = So, the problem reduces to that of just counting. We will talk more about counting techniques later. Equally likely outcomes Random sampling Tossing a fair coin Not equally likely outcomes Errors in different modules of a program Tossing an unfair coin Market ups and downs Ex: Suppose a dice is such that the even outcomes are twice as likely to occur as the odd outcomes; all the even outcomes are equally likely and all the odd outcomes are equally likely. O i = Outcome i occurs. (a) Find P(O i ) for i=1,2,3,4,5,6. (c) A = outcome is odd. Find P(A). (d) B = 3 outcome 5. Find P(B). 5
6 Relative frequency approach for assigning probability: Repeat the experiment a large number of times under identical conditions. # E = # times the event E occurs in n repetitions. P(E) = So, approximately, P(E) = long-term proportion of times E occurs. Ex: How to find P(It will rain tomorrow)? Subjective approach: If the experiment is either theoretically or practically unrepeatable, then use your personal degree of belief to assign probability. Note: Irrespective of which approach is used to assign probabilities, we generally use the relative frequency approach to interpret probabilities. Thus, P(E) is interpreted as the proportion of times the event A will occur in a large number of identical replications of the experiment. Ex: The experience of a computer manufacturer shows that 30% of the customers buy a flat screen monitor, 60% buy a printer and 20% buy both. (a) What is the probability that a customer will buy at least one of the two options? (b) What is the probability that a customer buys none of the two options? 6
CIVL Why are we studying probability and statistics? Learning Objectives. Basic Laws and Axioms of Probability
CIVL 3103 Basic Laws and Axioms of Probability Why are we studying probability and statistics? How can we quantify risks of decisions based on samples from a population? How should samples be selected
More informationLecture 1. Chapter 1. (Part I) Material Covered in This Lecture: Chapter 1, Chapter 2 ( ). 1. What is Statistics?
Lecture 1 (Part I) Material Covered in This Lecture: Chapter 1, Chapter 2 (2.1 --- 2.6). Chapter 1 1. What is Statistics? 2. Two definitions. (1). Population (2). Sample 3. The objective of statistics.
More informationProbability- describes the pattern of chance outcomes
Chapter 6 Probability the study of randomness Probability- describes the pattern of chance outcomes Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long
More informationLecture Lecture 5
Lecture 4 --- Lecture 5 A. Basic Concepts (4.1-4.2) 1. Experiment: A process of observing a phenomenon that has variation in its outcome. Examples: (E1). Rolling a die, (E2). Drawing a card form a shuffled
More informationENGI 4421 Introduction to Probability; Sets & Venn Diagrams Page α 2 θ 1 u 3. wear coat. θ 2 = warm u 2 = sweaty! θ 1 = cold u 3 = brrr!
ENGI 4421 Introduction to Probability; Sets & Venn Diagrams Page 2-01 Probability Decision trees u 1 u 2 α 2 θ 1 u 3 θ 2 u 4 Example 2.01 θ 1 = cold u 1 = snug! α 1 wear coat θ 2 = warm u 2 = sweaty! θ
More informationWeek 2. Section Texas A& M University. Department of Mathematics Texas A& M University, College Station 22 January-24 January 2019
Week 2 Section 1.2-1.4 Texas A& M University Department of Mathematics Texas A& M University, College Station 22 January-24 January 2019 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 1
More informationFundamentals of Probability CE 311S
Fundamentals of Probability CE 311S OUTLINE Review Elementary set theory Probability fundamentals: outcomes, sample spaces, events Outline ELEMENTARY SET THEORY Basic probability concepts can be cast in
More informationLecture notes for probability. Math 124
Lecture notes for probability Math 124 What is probability? Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments whose result
More informationChapter 6: Probability The Study of Randomness
Chapter 6: Probability The Study of Randomness 6.1 The Idea of Probability 6.2 Probability Models 6.3 General Probability Rules 1 Simple Question: If tossing a coin, what is the probability of the coin
More informationAxioms of Probability
Sample Space (denoted by S) The set of all possible outcomes of a random experiment is called the Sample Space of the experiment, and is denoted by S. Example 1.10 If the experiment consists of tossing
More informationChapter 2: Probability Part 1
Engineering Probability & Statistics (AGE 1150) Chapter 2: Probability Part 1 Dr. O. Phillips Agboola Sample Space (S) Experiment: is some procedure (or process) that we do and it results in an outcome.
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Lecture 2 Probability Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 3-1 3.1 Definition Random Experiment a process leading to an uncertain
More informationENGI 3423 Introduction to Probability; Sets & Venn Diagrams Page 3-01
ENGI 3423 Introduction to Probability; Sets & Venn Diagrams Page 3-01 Probability Decision trees θ 1 u 1 α 1 θ 2 u 2 Decision α 2 θ 1 u 3 Actions Chance nodes States of nature θ 2 u 4 Consequences; utility
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationProbability Theory and Applications
Probability Theory and Applications Videos of the topics covered in this manual are available at the following links: Lesson 4 Probability I http://faculty.citadel.edu/silver/ba205/online course/lesson
More informationSTAT Chapter 3: Probability
Basic Definitions STAT 515 --- Chapter 3: Probability Experiment: A process which leads to a single outcome (called a sample point) that cannot be predicted with certainty. Sample Space (of an experiment):
More informationSociology 6Z03 Topic 10: Probability (Part I)
Sociology 6Z03 Topic 10: Probability (Part I) John Fox McMaster University Fall 2014 John Fox (McMaster University) Soc 6Z03: Probability I Fall 2014 1 / 29 Outline: Probability (Part I) Introduction Probability
More informationCIVL 7012/8012. Basic Laws and Axioms of Probability
CIVL 7012/8012 Basic Laws and Axioms of Probability Why are we studying probability and statistics? How can we quantify risks of decisions based on samples from a population? How should samples be selected
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 3 Probability Contents 1. Events, Sample Spaces, and Probability 2. Unions and Intersections 3. Complementary Events 4. The Additive Rule and Mutually Exclusive
More informationMaking Hard Decision. Probability Basics. ENCE 627 Decision Analysis for Engineering
CHAPTER Duxbury Thomson Learning Making Hard Decision Probability asics Third Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering 7b FALL 003 y Dr. Ibrahim. Assakkaf
More informationThe probability of an event is viewed as a numerical measure of the chance that the event will occur.
Chapter 5 This chapter introduces probability to quantify randomness. Section 5.1: How Can Probability Quantify Randomness? The probability of an event is viewed as a numerical measure of the chance that
More informationCMPSCI 240: Reasoning about Uncertainty
CMPSCI 240: Reasoning about Uncertainty Lecture 2: Sets and Events Andrew McGregor University of Massachusetts Last Compiled: January 27, 2017 Outline 1 Recap 2 Experiments and Events 3 Probabilistic Models
More informationInformation Science 2
Information Science 2 Probability Theory: An Overview Week 12 College of Information Science and Engineering Ritsumeikan University Agenda Terms and concepts from Week 11 Basic concepts of probability
More informationIntroduction to Probability
Introduction to Probability Gambling at its core 16th century Cardano: Books on Games of Chance First systematic treatment of probability 17th century Chevalier de Mere posed a problem to his friend Pascal.
More informationChap 4 Probability p227 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a long series of
Chap 4 Probability p227 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a long series of repetitions. (p229) That is, probability is a long-term
More informationCIVL Probability vs. Statistics. Why are we studying probability and statistics? Basic Laws and Axioms of Probability
CIVL 3103 asic Laws and xioms of Probability Why are we studying probability and statistics? How can we quantify risks of decisions based on samples from a population? How should samples be selected to
More informationNotes Week 2 Chapter 3 Probability WEEK 2 page 1
Notes Week 2 Chapter 3 Probability WEEK 2 page 1 The sample space of an experiment, sometimes denoted S or in probability theory, is the set that consists of all possible elementary outcomes of that experiment
More informationSTT When trying to evaluate the likelihood of random events we are using following wording.
Introduction to Chapter 2. Probability. When trying to evaluate the likelihood of random events we are using following wording. Provide your own corresponding examples. Subjective probability. An individual
More informationProbability and Sample space
Probability and Sample space We call a phenomenon random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. The probability of any outcome
More informationLECTURE 1. 1 Introduction. 1.1 Sample spaces and events
LECTURE 1 1 Introduction The first part of our adventure is a highly selective review of probability theory, focusing especially on things that are most useful in statistics. 1.1 Sample spaces and events
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions
Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions 1999 Prentice-Hall, Inc. Chap. 4-1 Chapter Topics Basic Probability Concepts: Sample
More informationSTAT 430/510 Probability
STAT 430/510 Probability Hui Nie Lecture 3 May 28th, 2009 Review We have discussed counting techniques in Chapter 1. Introduce the concept of the probability of an event. Compute probabilities in certain
More informationELEG 3143 Probability & Stochastic Process Ch. 1 Probability
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 1 Probability Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Applications Elementary Set Theory Random
More informationRecap. The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY INFERENTIAL STATISTICS
Recap. Probability (section 1.1) The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY Population Sample INFERENTIAL STATISTICS Today. Formulation
More informationProbability COMP 245 STATISTICS. Dr N A Heard. 1 Sample Spaces and Events Sample Spaces Events Combinations of Events...
Probability COMP 245 STATISTICS Dr N A Heard Contents Sample Spaces and Events. Sample Spaces........................................2 Events........................................... 2.3 Combinations
More informationThe enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail}
Random Experiment In random experiments, the result is unpredictable, unknown prior to its conduct, and can be one of several choices. Examples: The Experiment of tossing a coin (head, tail) The Experiment
More informationMATH2206 Prob Stat/20.Jan Weekly Review 1-2
MATH2206 Prob Stat/20.Jan.2017 Weekly Review 1-2 This week I explained the idea behind the formula of the well-known statistic standard deviation so that it is clear now why it is a measure of dispersion
More informationReview of Basic Probability
Review of Basic Probability Erik G. Learned-Miller Department of Computer Science University of Massachusetts, Amherst Amherst, MA 01003 September 16, 2009 Abstract This document reviews basic discrete
More informationSTP 226 ELEMENTARY STATISTICS
STP 226 ELEMENTARY STATISTICS CHAPTER 5 Probability Theory - science of uncertainty 5.1 Probability Basics Equal-Likelihood Model Suppose an experiment has N possible outcomes, all equally likely. Then
More informationACM 116: Lecture 1. Agenda. Philosophy of the Course. Definition of probabilities. Equally likely outcomes. Elements of combinatorics
1 ACM 116: Lecture 1 Agenda Philosophy of the Course Definition of probabilities Equally likely outcomes Elements of combinatorics Conditional probabilities 2 Philosophy of the Course Probability is the
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14 Introduction One of the key properties of coin flips is independence: if you flip a fair coin ten times and get ten
More information3.2 Probability Rules
3.2 Probability Rules The idea of probability rests on the fact that chance behavior is predictable in the long run. In the last section, we used simulation to imitate chance behavior. Do we always need
More informationChapter 5 : Probability. Exercise Sheet. SHilal. 1 P a g e
1 P a g e experiment ( observing / measuring ) outcomes = results sample space = set of all outcomes events = subset of outcomes If we collect all outcomes we are forming a sample space If we collect some
More informationPreliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix
More informationMean, Median and Mode. Lecture 3 - Axioms of Probability. Where do they come from? Graphically. We start with a set of 21 numbers, Sta102 / BME102
Mean, Median and Mode Lecture 3 - Axioms of Probability Sta102 / BME102 Colin Rundel September 1, 2014 We start with a set of 21 numbers, ## [1] -2.2-1.6-1.0-0.5-0.4-0.3-0.2 0.1 0.1 0.2 0.4 ## [12] 0.4
More informationMATH MW Elementary Probability Course Notes Part I: Models and Counting
MATH 2030 3.00MW Elementary Probability Course Notes Part I: Models and Counting Tom Salisbury salt@yorku.ca York University Winter 2010 Introduction [Jan 5] Probability: the mathematics used for Statistics
More informationProbability: Sets, Sample Spaces, Events
Probability: Sets, Sample Spaces, Events Engineering Statistics Section 2.1 Josh Engwer TTU 01 February 2016 Josh Engwer (TTU) Probability: Sets, Sample Spaces, Events 01 February 2016 1 / 29 The Need
More informationBusiness Statistics. Lecture 3: Random Variables and the Normal Distribution
Business Statistics Lecture 3: Random Variables and the Normal Distribution 1 Goals for this Lecture A little bit of probability Random variables The normal distribution 2 Probability vs. Statistics Probability:
More informationExample: Suppose we toss a quarter and observe whether it falls heads or tails, recording the result as 1 for heads and 0 for tails.
Example: Suppose we toss a quarter and observe whether it falls heads or tails, recording the result as 1 for heads and 0 for tails. (In Mathematical language, the result of our toss is a random variable,
More informationI - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability
What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)
More information18.600: Lecture 3 What is probability?
18.600: Lecture 3 What is probability? Scott Sheffield MIT Outline Formalizing probability Sample space DeMorgan s laws Axioms of probability Outline Formalizing probability Sample space DeMorgan s laws
More informationBasic Statistics and Probability Chapter 3: Probability
Basic Statistics and Probability Chapter 3: Probability Events, Sample Spaces and Probability Unions and Intersections Complementary Events Additive Rule. Mutually Exclusive Events Conditional Probability
More informationCS626 Data Analysis and Simulation
CS626 Data Analysis and Simulation Instructor: Peter Kemper R 104A, phone 221-3462, email:kemper@cs.wm.edu Today: Probability Primer Quick Reference: Sheldon Ross: Introduction to Probability Models 9th
More informationAnnouncements. Topics: To Do:
Announcements Topics: In the Probability and Statistics module: - Sections 1 + 2: Introduction to Stochastic Models - Section 3: Basics of Probability Theory - Section 4: Conditional Probability; Law of
More informationLecture 1 : The Mathematical Theory of Probability
Lecture 1 : The Mathematical Theory of Probability 0/ 30 1. Introduction Today we will do 2.1 and 2.2. We will skip Chapter 1. We all have an intuitive notion of probability. Let s see. What is the probability
More informationIntroduction to Probability
Introduction to Probability Content Experiments, Counting Rules, and Assigning Probabilities Events and Their Probability Some Basic Relationships of Probability Conditional Probability Bayes Theorem 2
More informationChapter 4 - Introduction to Probability
Chapter 4 - Introduction to Probability Probability is a numerical measure of the likelihood that an event will occur. Probability values are always assigned on a scale from 0 to 1. A probability near
More information4. Probability of an event A for equally likely outcomes:
University of California, Los Angeles Department of Statistics Statistics 110A Instructor: Nicolas Christou Probability Probability: A measure of the chance that something will occur. 1. Random experiment:
More informationChapter 1: Revie of Calculus and Probability
Chapter 1: Revie of Calculus and Probability Refer to Text Book: Operations Research: Applications and Algorithms By Wayne L. Winston,Ch. 12 Operations Research: An Introduction By Hamdi Taha, Ch. 12 OR441-Dr.Khalid
More informationSample Space: Specify all possible outcomes from an experiment. Event: Specify a particular outcome or combination of outcomes.
Chapter 2 Introduction to Probability 2.1 Probability Model Probability concerns about the chance of observing certain outcome resulting from an experiment. However, since chance is an abstraction of something
More informationStatistics 251: Statistical Methods
Statistics 251: Statistical Methods Probability Module 3 2018 file:///volumes/users/r/renaes/documents/classes/lectures/251301/renae/markdown/master%20versions/module3.html#1 1/33 Terminology probability:
More informationTOPIC 12 PROBABILITY SCHEMATIC DIAGRAM
TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of Importance References NCERT Book Vol. II Probability (i) Conditional Probability *** Article 1.2 and 1.2.1 Solved Examples 1 to 6 Q. Nos
More informationF71SM STATISTICAL METHODS
F71SM STATISTICAL METHODS RJG SUMMARY NOTES 2 PROBABILITY 2.1 Introduction A random experiment is an experiment which is repeatable under identical conditions, and for which, at each repetition, the outcome
More informationStats Probability Theory
Stats 241.3 Probability Theory Instructor: Office: W.H.Laverty 235 McLean Hall Phone: 966-6096 Lectures: Evaluation: M T W Th F 1:30pm - 2:50pm Thorv 105 Lab: T W Th 3:00-3:50 Thorv 105 Assignments, Labs,
More informationMAE 493G, CpE 493M, Mobile Robotics. 6. Basic Probability
MAE 493G, CpE 493M, Mobile Robotics 6. Basic Probability Instructor: Yu Gu, Fall 2013 Uncertainties in Robotics Robot environments are inherently unpredictable; Sensors and data acquisition systems are
More informationProbability Rules. MATH 130, Elements of Statistics I. J. Robert Buchanan. Fall Department of Mathematics
Probability Rules MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Introduction Probability is a measure of the likelihood of the occurrence of a certain behavior
More informationSTA Module 4 Probability Concepts. Rev.F08 1
STA 2023 Module 4 Probability Concepts Rev.F08 1 Learning Objectives Upon completing this module, you should be able to: 1. Compute probabilities for experiments having equally likely outcomes. 2. Interpret
More informationProbability: Axioms, Properties, Interpretations
Probability: Axioms, Properties, Interpretations Engineering Statistics Section 2.2 Josh Engwer TTU 03 February 2016 Josh Engwer (TTU) Probability: Axioms, Properties, Interpretations 03 February 2016
More informationIntroduction to probability
Introduction to probability 4.1 The Basics of Probability Probability The chance that a particular event will occur The probability value will be in the range 0 to 1 Experiment A process that produces
More informationAP Statistics Ch 6 Probability: The Study of Randomness
Ch 6.1 The Idea of Probability Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. We call a phenomenon random if individual outcomes are uncertain
More informationStatistical Inference
Statistical Inference Lecture 1: Probability Theory MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 11, 2018 Outline Introduction Set Theory Basics of Probability Theory
More informationProbability deals with modeling of random phenomena (phenomena or experiments whose outcomes may vary)
Chapter 14 From Randomness to Probability How to measure a likelihood of an event? How likely is it to answer correctly one out of two true-false questions on a quiz? Is it more, less, or equally likely
More informationMathematical Probability
Mathematical Probability STA 281 Fall 2011 1 Introduction Engineers and scientists are always exposed to data, both in their professional capacities and in everyday activities. The discipline of statistics
More informationBasic notions of probability theory
Basic notions of probability theory Contents o Boolean Logic o Definitions of probability o Probability laws Why a Lecture on Probability? Lecture 1, Slide 22: Basic Definitions Definitions: experiment,
More information4 Lecture 4 Notes: Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio
4 Lecture 4 Notes: Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio Wrong is right. Thelonious Monk 4.1 Three Definitions of
More informationToday we ll discuss ways to learn how to think about events that are influenced by chance.
Overview Today we ll discuss ways to learn how to think about events that are influenced by chance. Basic probability: cards, coins and dice Definitions and rules: mutually exclusive events and independent
More informationUNIT 5 ~ Probability: What Are the Chances? 1
UNIT 5 ~ Probability: What Are the Chances? 1 6.1: Simulation Simulation: The of chance behavior, based on a that accurately reflects the phenomenon under consideration. (ex 1) Suppose we are interested
More informationProbability. 25 th September lecture based on Hogg Tanis Zimmerman: Probability and Statistical Inference (9th ed.)
Probability 25 th September 2017 lecture based on Hogg Tanis Zimmerman: Probability and Statistical Inference (9th ed.) Properties of Probability Methods of Enumeration Conditional Probability Independent
More informationLecture 1. ABC of Probability
Math 408 - Mathematical Statistics Lecture 1. ABC of Probability January 16, 2013 Konstantin Zuev (USC) Math 408, Lecture 1 January 16, 2013 1 / 9 Agenda Sample Spaces Realizations, Events Axioms of Probability
More informationLecture 3 - Axioms of Probability
Lecture 3 - Axioms of Probability Sta102 / BME102 January 25, 2016 Colin Rundel Axioms of Probability What does it mean to say that: The probability of flipping a coin and getting heads is 1/2? 3 What
More informationProbability the chance that an uncertain event will occur (always between 0 and 1)
Quantitative Methods 2013 1 Probability as a Numerical Measure of the Likelihood of Occurrence Probability the chance that an uncertain event will occur (always between 0 and 1) Increasing Likelihood of
More informationEvent A: at least one tail observed A:
Chapter 3 Probability 3.1 Events, sample space, and probability Basic definitions: An is an act of observation that leads to a single outcome that cannot be predicted with certainty. A (or simple event)
More informationOrigins of Probability Theory
1 16.584: INTRODUCTION Theory and Tools of Probability required to analyze and design systems subject to uncertain outcomes/unpredictability/randomness. Such systems more generally referred to as Experiments.
More informationSection 13.3 Probability
288 Section 13.3 Probability Probability is a measure of how likely an event will occur. When the weather forecaster says that there will be a 50% chance of rain this afternoon, the probability that it
More informationHomework (due Wed, Oct 27) Chapter 7: #17, 27, 28 Announcements: Midterm exams keys on web. (For a few hours the answer to MC#1 was incorrect on
Homework (due Wed, Oct 27) Chapter 7: #17, 27, 28 Announcements: Midterm exams keys on web. (For a few hours the answer to MC#1 was incorrect on Version A.) No grade disputes now. Will have a chance to
More informationWith Question/Answer Animations. Chapter 7
With Question/Answer Animations Chapter 7 Chapter Summary Introduction to Discrete Probability Probability Theory Bayes Theorem Section 7.1 Section Summary Finite Probability Probabilities of Complements
More informationAMS7: WEEK 2. CLASS 2
AMS7: WEEK 2. CLASS 2 Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio Friday April 10, 2015 Probability: Introduction Probability:
More informationCompound Events. The event E = E c (the complement of E) is the event consisting of those outcomes which are not in E.
Compound Events Because we are using the framework of set theory to analyze probability, we can use unions, intersections and complements to break complex events into compositions of events for which it
More informationExperiment -- the process by which an observation is made. Sample Space -- ( S) the collection of ALL possible outcomes of an experiment
A. 1 Elementary Probability Set Theory Experiment -- the process by which an observation is made Ex. Outcome The result of a chance experiment. Ex. Sample Space -- ( S) the collection of ALL possible outcomes
More informationtossing a coin selecting a card from a deck measuring the commuting time on a particular morning
2 Probability Experiment An experiment or random variable is any activity whose outcome is unknown or random upfront: tossing a coin selecting a card from a deck measuring the commuting time on a particular
More informationMath 1313 Experiments, Events and Sample Spaces
Math 1313 Experiments, Events and Sample Spaces At the end of this recording, you should be able to define and use the basic terminology used in defining experiments. Terminology The next main topic in
More informationSixth Edition. Chapter 2 Probability. Copyright 2014 John Wiley & Sons, Inc. All rights reserved. Probability
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 2 Probability 2 Probability CHAPTER OUTLINE 2-1 Sample Spaces and Events 2-1.1 Random Experiments
More informationBASICS OF PROBABILITY CHAPTER-1 CS6015-LINEAR ALGEBRA AND RANDOM PROCESSES
BASICS OF PROBABILITY CHAPTER-1 CS6015-LINEAR ALGEBRA AND RANDOM PROCESSES COMMON TERMS RELATED TO PROBABILITY Probability is the measure of the likelihood that an event will occur Probability values are
More informationExample. What is the sample space for flipping a fair coin? Rolling a 6-sided die? Find the event E where E = {x x has exactly one head}
Chapter 7 Notes 1 (c) Epstein, 2013 CHAPTER 7: PROBABILITY 7.1: Experiments, Sample Spaces and Events Chapter 7 Notes 2 (c) Epstein, 2013 What is the sample space for flipping a fair coin three times?
More informationProbability. Chapter 1 Probability. A Simple Example. Sample Space and Probability. Sample Space and Event. Sample Space (Two Dice) Probability
Probability Chapter 1 Probability 1.1 asic Concepts researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people
More informationBasic notions of probability theory
Basic notions of probability theory Contents o Boolean Logic o Definitions of probability o Probability laws Objectives of This Lecture What do we intend for probability in the context of RAM and risk
More informationTopic -2. Probability. Larson & Farber, Elementary Statistics: Picturing the World, 3e 1
Topic -2 Probability Larson & Farber, Elementary Statistics: Picturing the World, 3e 1 Probability Experiments Experiment : An experiment is an act that can be repeated under given condition. Rolling a
More informationLecture Notes 1 Basic Probability. Elements of Probability. Conditional probability. Sequential Calculation of Probability
Lecture Notes 1 Basic Probability Set Theory Elements of Probability Conditional probability Sequential Calculation of Probability Total Probability and Bayes Rule Independence Counting EE 178/278A: Basic
More informationChapter 5: Probability in Our Daily Lives
Chapter 5: in Our Daily Lives These notes reflect material from our text, Statistics: The Art and Science of Learning from Data, Third Edition, by Alan Agresti and Catherine Franklin, published by Pearson,
More informationan event with one outcome is called a simple event.
Ch5Probability Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations
More information